Visualizing two qubits

The notions of entanglement witnesses and separable and entangled states for a two qubit system can be visualized in three dimensions using the SLOCC equivalence classes. This visualization preserves the duality relations between the various sets and allows us to give “proof by inspection” of a nonelementary result of Horodecki et al. [Phys. Lett. A 223, 1–8 (1996)] that for two qubits, Peres PPT (positive partial transpose) test for separability is iff. We then show that the CHSH Bell inequalities can be visualized as circles and cylinders in the same diagram. This allows us to give a geometric proof of yet another result of Horodecki et al. [Phys. Lett. A 200, 340–344 (1995)], which optimizes the violation of the CHSH Bell inequality. Finally, we give numerical evidence that, remarkably, allowing Alice and Bob to use three rather than two measurements each does not help them to distinguish any new entangled SLOCC equivalence class beyond the CHSH class.